Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs

Consider the restriction of an irreducible unitary representation p of a Lie group G to its subgroup H. Kirillov's revolutionary idea on the orbit method suggests that the multiplicity of an irreducible H-module. occurring in the restriction p| H could be read from the coadjoint action of H on OG n pr -1(OH), provided p and. are ` geometric quantizations' of a G-coadjoint orbit OG and an H-coadjoint orbit OH, respectively, where pr : v -1g *. v -1h * is the projection dual to the inclusion h. g of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits OG of a semisimple Lie group G corresponding to highest weight modules of scalar type. We prove that the CorwinGreenleaf number (OG n pr -1(OH))/ H is either zero or one for any H-coadjoint orbit OH, whenever (G, H) is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits OH with nonzero Corwin-Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as ` classical limits' of the multiplicity-free branching laws of holomorphic discrete series representations (Kobayashi [ Progr. Math. 2007]).